Mathematics for the interested outsider

Proving the Classification Theorem IV

Any connected graph takes one of the four following forms: a simple chain, the graph, three simple chains joined at a central vertex, or a chain with exactly one double edge.

This step largely consolidates what we’ve done to this point. Here are the four possible graphs:

The labels will help with later steps.

Step 5 told us that there’s only one connected graph that contains a triple edge. Similarly, if we had more than one double edge or triple vertex, then we must be able to find two of them connected by a simple chain. But that will violate step 7, and so we can only have one of these features either.

The only possible Coxeter graphs with a double edge are those underlying the Dynkin diagrams , , and .

Here we’ll use the labels on the above graph. We define

As in step 6, we find that and all other pairs of vectors are orthogonal. And so we calculate

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